Electromagnetic induction logging instruments are used in the oil and gas industry to measure conductivity of geological formations penetrated by wellbores. A wellbore includes the openhole or uncased portion of an oil or gas well. A borehole may refer to the inside diameter of the wellbore wall, the rock face that bounds the drilled hole. The electromagnetic induction logging technology was first introduced in 1949, and more fully described by Henri G. Doll, “Introduction to Induction Logging and Application to Logging of Wells Drilled with Oil Based Mud,” Journal of Petroleum Technology, Vol. 1, No. 6, June 1949, pp. 148–162.
In induction well logging, a coil system is lowered into a wellbore for the purpose of investigating the electrical properties of earth formations adjacent the wellbore. An electrical property of interest in such investigation is the electrical conductivity of particular portions of the formation.
Electromagnetic induction logging instruments, or tools, typically consist of at least one transmitting coil and at least one receiving coil. An alternating current having at least one frequency is conducted through the transmitting coil(s). The alternating current induces eddy currents to flow within the surrounding geological formations. The eddy current in turn induces voltages in the receiving coil(s). The voltages induced in the receiving coil(s) are converted to apparent conductivities through a linear relationship derived under the assumption of a vanishingly small value of conductivity. This assumption is commonly referred to as “the Doll limit.” When the formation conductivity value is not vanishingly small, the voltages induced in the receiver coil(s) relate non-linearly to the formation conductivity. The difference between the derived linear relationship and the actual nonlinear formation conductivity is generically referred to as the “skin effect.” There are two important aspects of this skin effect. First, the skin effect makes the apparent conductivity value smaller than its true value. Second, the skin effect makes the response functions or geometric factors different from the ones derived under the Doll limit, thus causing severe non-linearity of the measured values.
Induction logging measurements are affected by skin effect in the aspects of apparent conductivity value and in the aspect of response functions or geometric factors as well, causing severe non-linearity. Traditional skin effect correction methods only boost the apparent conductivity and are no longer suited for modern induction tools, such as array induction tools, which rely on linear numerical processing to obtain desired induction logs. The skin effect correction method with quadrature (x-signal) measurements is widely used at present. However, the quadrature signal component is prone to various noise, error, instability, etc., from the direct coupling through the geological formation. In order to minimize interference during quadrature signal measurements, a metal mandrel is generally required in modern induction tools to provide adequate electromagnetic shielding between the transmitter coil lead wires and the receiver coils and receiver coil lead wires. A metal mandrel blocks the induction signal all through its cross section, thus for adequate signal level, the induction tool must have a large diameter, making it almost impossible to design a slim-hole array induction tool.
Over the years, several different methods were developed to correct for the skin effect. The traditional method, used with the induction tools that measure only the in-phase part of the induced voltage with a single frequency, boosts the apparent conductivity by empirical sequence or formulas. This traditional method is in principle only a stopgap to the skin effect, because it only addresses the first aspect of the skin effect. A similar method is also published by Liu et al. in paper “A New Method to Correct the Effect of Skin-Effect in Induction Logs,” SPWLA 41st Annual Logging Symposium, Jun. 4–7, 2000, paper D.
U.S. Pat. No. 3,147,429, by Moran (Moran '429), describes a skin effect correction method of combining the in-phase signal component and the out-of-phase (in quadrature) signal component. This method addresses both aspects of the skin effect. However, the measurement of the quadrature signal component often suffers from relatively small signal levels (when the formation is resistive), sonde error instability, severe temperature effects and distortion from magnetic materials in the drilling fluid. The aforementioned drawbacks of measuring the quadrature signal component eventually limit the skin effect correction. U.S. Pat. No. 4,471,436, by Schaefer et al., and U.S. Pat. No. 5,698,982, by Mitchell describe different implementations of the method described in the aforementioned Moran '429.
To avoid having to use the quadrature signal component, two methods were developed in the past as described in U.S. Pat. No. 5,146,167, by Strickland et al. (Strickland '167), and U.S. Pat. No. 5,666,057, by Beard et al. (Beard '057). Strickland '167 describes a skin effect correction method that uses only the in-phase signal component. By filtering the cubed square root of the in-phase signal component, this method corrects for distortion of the vertical geometric factors while restoring the conductivity value. But, it leaves the radial geometric factor uncorrected. The method described in Beard '057 first determines a best fit curve of the in-phase measurements with respect to the frequency and then extracts the first derivative and the second derivative of the best fit curve with respect to the frequency. The first and second derivatives are then used to extrapolate the curve of the in-phase measurement with respect to a zero frequency. The skin effect corrected conductivity is the conductivity that would be obtained when the frequency is equal to zero. To apply the method of Beard '057, a set of in-phase measurements must be made at multiple frequencies to form a simple spectrum with respect to frequency, which increases the cost of the induction logging.